133 research outputs found
Homological Pisot Substitutions and Exact Regularity
We consider one-dimensional substitution tiling spaces where the dilatation
(stretching factor) is a degree d Pisot number, and where the first rational
Cech cohomology is d-dimensional. We construct examples of such "homological
Pisot" substitutions that do not have pure discrete spectra. These examples are
not unimodular, and we conjecture that the coincidence rank must always divide
a power of the norm of the dilatation. To support this conjecture, we show that
homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in
which the number of occurrences of a patch for a return length is governed
strictly by the length. The ERP puts strong constraints on the measure of any
cylinder set in the corresponding tiling space.Comment: 16 pages, LaTeX, no figure
Symmetric intersections of Rauzy fractals
In this article we study symmetric subsets of Rauzy fractals of unimodular
irreducible Pisot substitutions. The symmetry considered is reflection through
the origin. Given an unimodular irreducible Pisot substitution, we consider the
intersection of its Rauzy fractal with the Rauzy fractal of the reverse
substitution. This set is symmetric and it is obtained by the balanced pair
algorithm associated with both substitutions
The geometry of non-unit Pisot substitutions
Let be a non-unit Pisot substitution and let be the
associated Pisot number. It is known that one can associate certain fractal
tiles, so-called \emph{Rauzy fractals}, with . In our setting, these
fractals are subsets of a certain open subring of the ad\`ele ring
. We present several approaches on how to
define Rauzy fractals and discuss the relations between them. In particular, we
consider Rauzy fractals as the natural geometric objects of certain numeration
systems, define them in terms of the one-dimensional realization of
and its dual (in the spirit of Arnoux and Ito), and view them as the dual of
multi-component model sets for particular cut and project schemes. We also
define stepped surfaces suited for non-unit Pisot substitutions. We provide
basic topological and geometric properties of Rauzy fractals associated with
non-unit Pisot substitutions, prove some tiling results for them, and provide
relations to subshifts defined in terms of the periodic points of , to
adic transformations, and a domain exchange. We illustrate our results by
examples on two and three letter substitutions.Comment: 29 page
Pisot conjecture and Rauzy fractals
We provide a proof of Pisot conjecture, a classification problem in Ergodic
Theory on recurrent sequences generated by irreducible Pisot substitutions.Comment: revise
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